σ-Entropy Structures and IrreversibilityFrom Algebraic Collapse to Thermodynamic Entropy

Abstract

We introduce σ-entropy structures, extending the saturation-collapse framework originally
developed by Manafi (2026). Building on the σ-filtered dynamical systems, where admissible
endomorphisms non-decrease saturation and irreversibility emerges from endomorphism degeneration
beyond a critical threshold, we incorporate additivity for composite systems and a
probabilistic structure over measurable distributions p, defining averaged saturation.
This extension endows σ with entropy-like properties, yielding rigorous theorems on monotonicity
under irreversible pushforwards (Second-Law analog), non-negative entropy production
with equality precisely for reversible maps, an σ-H-theorem generalizing Boltzmann’s result,
fluctuation inequalities, σ-Boltzmann entropy for macrostates, and a variational principle recovering
Gibbs-like distributions.
Classical entropies in dynamical systems and thermodynamics arise as special cases (e.g.,
σ(x) = −log ρ(x) for probability density ρ). The resulting framework derives thermodynamic
irreversibility and entropy growth endogenously from algebraic collapse, providing a unified
bridge between threshold phenomena in abstract algebra and nonequilibrium physics.
Keywords: saturation-collapse, σ-entropy structures, irreversibility, algebraic collapse, entropy
production, σ-H-theorem, thermodynamic entropy, endomorphism degeneration, partial
algebraic structures, nonequilibrium thermodynamics